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Poisson multi-Bernoulli mixture filter: direct derivation and implementation

Ángel F. García-Fernández, Jason L. Williams, Karl Granström, Lennart Svensson

arXiv:1703.04264v4cs.CVstat.ME

TL;DR

The paper tackles the accessibility and implementation of PMBM filtering for standard point-target multi-target tracking. It provides a non-PGFL derivation, relates PMBM to δ-GLMB, and develops linear/Gaussian implementation and estimator procedures. The PMBM implementation is compared with other RFS filters in a challenging scenario and is reported to outperform them.

  • Problem

    PMBM filtering had previously been derived using PGFLs and functional derivatives, limiting accessibility of its theoretical structure and practical implementation.

  • Method

    The paper derives PMBM without PGFLs, interprets δ-GLMB as a labelled special case, implements PMBM for linear/Gaussian models, and obtains three estimators.

  • Results

    The PMBM filter is reported to outperform other multi-target filters in a challenging scenario.

  • Takeaways & Limitations

    The PMBM parameterisation offers a more efficient filtering-density representation than δ-GLMB, simplifying prediction and supporting efficient estimator extraction.

Abstract

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We provide a derivation of the Poisson multi-Bernoulli mixture (PMBM) filter for multi-target tracking with the standard point target measurements without using probability generating functionals or functional derivatives. We also establish the connection with the δ-generalised labelled multi-Bernoulli (δ-GLMB) filter, showing that a δ-GLMB density represents a multi-Bernoulli mixture with labelled targets so it can be seen as a special case of PMBM. In addition, we propose an implementation for linear/Gaussian dynamic and measurement models and how to efficiently obtain typical estimators in the literature from the PMBM. The PMBM filter is shown to outperform other filters in the literature in a challenging scenario.

I. INTRODUCTION

The paper addresses intractable Bayesian multi-target tracking by making the PMBM filter more accessible through a non-PGFL derivation, implementation, and estimator procedures.

  • MTT infers the states of an unknown, changing number of targets from noisy measurements in applications including surveillance and autonomous driving.
  • Bayesian RFS filtering is generally intractable, motivating computationally expensive approximations such as particle filters.
  • Conjugate priors keep posterior densities within the same family and can express updates through single-target Bayesian updates.
  • For standard point-target measurements, PMBM combines a Poisson process with a multi-Bernoulli mixture representing data-association hypotheses.
  • The paper derives PMBM without PGFLs or functional derivatives, connects it to δ-GLMB, proposes a linear/Gaussian implementation, and develops estimators.
  • B. Standard point target measurement model: The standard measurement model decomposes measurements into independent clutter and target-generated sets, with each target producing zero or one measurement.

III. PROOF OF THE CONJUGACY OF THE PMBM

The proof represents the PMBM prior as a Poisson component joined with a multi-Bernoulli mixture, then explains how measurements generate and update association hypotheses.

  • The derivation uses alternative likelihood representations and component independence to establish conjugacy without relying on PGFLs.
  • The Poisson component models undetected targets, while each potentially detected target is represented by a Bernoulli component.
  • Global hypotheses encode possible associations of measurements to potentially detected targets, with single-target hypotheses recording association histories.
  • The conjugate prior is the union of a Poisson density and a multi-Bernoulli mixture.
  • A new Bernoulli component is created for each measurement; existence represents a first detection, whereas non-existence represents clutter or reassignment to another target.

B. Update of a Poisson prior

The Poisson-prior update decomposes the posterior into an undetected-target Poisson process and Bernoulli components associated with the measurements.

  • The likelihood is decomposed by partitioning targets into undetected targets and measurement-origin sets, each containing one target or being empty for clutter.
  • After observing Z, the Poisson posterior is a union of a Poisson process and a multi-Bernoulli RFS.
  • The updated Poisson intensity equals the prior intensity multiplied by 1 − pd(·).
  • Each measurement produces a Bernoulli component whose existence probability represents whether it originated from a target rather than clutter or another association.

2) Update of one Bernoulli component:

The Bernoulli update considers the only measurement subsets with nonzero likelihood: a singleton measurement or no measurement. These cases produce explicit Bernoulli posteriors that feed the PMBM update.

  • The update evaluates Bernoulli component fj,i (·) using measurement subset Zi and likelihood t (Zi|·).
  • For Zi = {z}, the updated Bernoulli has existence probability 1 and state density proportional to pd (x) l (z|x) pj,i (x).
  • For Zi = ∅, the likelihood can be nonzero when Xi is either {x} or ∅.
  • In the no-measurement case, the updated state density is proportional to (1 −pd (x)) pj,i (x).
  • The conjugate-prior update assigns zero existence to new Bernoulli components associated with measurements excluded from Zy.

D. Prediction of the conjugate prior

The prediction step preserves the PMBM structure under the standard dynamic and measurement models. Its Poisson and multi-Bernoulli-mixture parameters are propagated using established Poisson and MeMBer prediction equations.

  • The predicted Poisson intensity is obtained using the PHD filter prediction equation.
  • Predicted multi-Bernoulli-mixture parameters follow the multi-target multi-Bernoulli filter prediction equation.
  • Standard models represent detection and survival with pd (·) and ps (·), target measurements and births with l (·|·) and g (·|·), and clutter and new births as Poisson processes.
  • The derivation uses independent RFS components and set-valued-function identities to obtain the predicted density.
  • The density of the measurement is the union of a Poisson process and a multi-Bernoulli mixture with the prior weights and specified updated parameters.

E. Conjugacy for multi-Bernoulli mixtures

The paper establishes conjugacy for multi-Bernoulli mixtures and labelled variants under standard point-target dynamics and measurements. It also explains the δ-GLMB connection through labelled MBM representations and the MBM01 parameterisation.

  • E. Conjugacy for multi-Bernoulli mixtures: The MBM family is a conjugate prior when the birth process is multi-Bernoulli or an MBM.
  • E. Conjugacy for multi-Bernoulli mixtures: For multi-Bernoulli birth, prediction adds Bernoulli components to each mixture term; for MBM birth, it combines every old and birth mixture-term pair.
  • Connection with δ-GLMB: The δ-GLMB density is presented as a labelled multi-Bernoulli mixture with a less efficient storage and computational parameterisation.
  • A. Multi-Bernoulli mixture 01 parameterisation: A Bernoulli with non-deterministic existence can be expanded into components whose existence probabilities are either zero or one.
  • A. Multi-Bernoulli mixture 01 parameterisation: The MBM01 representation may cause a tremendous increase in mixture components, making it inefficient and increasing brute-force computational complexity.
  • Labelled multi-Bernoulli mixtures: Labels augment target states with unique, time-fixed identifiers, yielding labelled MBMs whose densities retain the conjugacy property.

C. Relation between δ-GLMB densities and labelled multi-Bernoulli mixtures

The paper establishes that δ-GLMB densities and labelled MBM01 representations encode the same labelled multi-target densities, while PMBM/MBM parameterisations offer computational advantages through more flexible mixture representations.

  • δ-GLMB densities and labelled MBM01 parameterisations can represent the same labelled multi-target densities with the same number of global hypotheses.
  • Both representations use labelled targets and global hypotheses with deterministic target existence.
  • Unlike δ-GLMB notation, MBM01 can represent both labelled and unlabelled targets.
  • With multi-Bernoulli births and survival probability below one, MBM avoids the 2^n global-hypothesis expansion produced by MBM01/δ-GLMB representations.
  • One MBM global hypothesis can efficiently represent many δ-GLMB global hypotheses, simplifying prediction and update independently of label usage.
  • For Poisson births, PMBM represents the Poisson component by its intensity, whereas labelled δ-GLMB would require infinitely many global hypotheses.

C. Update

The PMBM update creates hypotheses for undetected, newly detected, and previously detected targets, then forms global data-association hypotheses and prunes them for tractable computation.

  • The update handles undetected targets, first-time detections, and previously potentially detected targets as three distinct cases.
  • Ellipsoidal gating first limits measurements considered for new Bernoulli components created from the Poisson prior.
  • For previously detected targets, the filter creates a misdetection hypothesis and gated detection hypotheses with updated weights and Bernoulli parameters.
  • Each previous global hypothesis generates updated global hypotheses by assigning every gated measurement to an existing or new track.
  • Murty’s algorithm selects the k highest-weight global hypotheses without evaluating every newly generated association.
  • Algorithm 1 performs prediction, gated updates, hypothesis generation, Murty selection, state estimation, and pruning.

VI. ESTIMATION

The PMBM supports tractable estimators based on either the highest-weight global hypothesis or the maximum a posteriori cardinality, with target means extracted from selected Bernoulli components.

  • Estimator 1 selects the highest-weight multi-Bernoulli mixture hypothesis and reports means of Bernoulli components whose existence exceeds a threshold.
  • The existence threshold determines how many consecutive target misdetections can occur before an estimate is no longer reported.
  • Estimator 2 first finds the MAP cardinality, then selects the highest-weight global hypothesis having that cardinality.
  • The MBM cardinality distribution is computed efficiently using a discrete Fourier transform because Bernoulli cardinality distributions convolve.
  • After selecting the cardinality-constrained hypothesis, the estimate uses the means of its Bernoulli components with the highest existence probabilities.

C. Estimator 3

Estimator 3 selects the highest-weight global hypothesis with deterministic cardinality and reports the means of its existing Bernoulli components. In the challenging simulation, PMBM-based estimators performed strongly, with Estimator 1 lowest and Estimators 2 and 3 similar.

  • C. Estimator 3: Estimator 3 selects the global hypothesis with deterministic cardinality having the highest weight.Its deterministic cardinality is obtained by expanding each Bernoulli component into existence and non-existence alternatives.
  • C. Estimator 3: The resulting set estimate reports the means of Bernoulli components whose existence probabilities exceed 0.5.
  • VII. Simulations: 81 time steps and 100 Monte Carlo runs were used to evaluate root mean square OSPA error in a four-target scenario with clutter and target proximity.The scenario includes a target death at time step 40 while targets are in close proximity.
  • VII. Simulations: Estimator 1 applied to PMBM provides the lowest errors, while Estimators 2 and 3 behave similarly.MOMB performs as accurately as PMBM Estimators 2 and 3 in this scenario.
  • VII. Simulations: Across detection probabilities, PMBM performs better than the other filters regardless of estimator.Estimator 1 is better for pd equal to or above 0.9, whereas Estimator 2 is best for lower pd values.
  • VIII. Conclusions: The paper concludes that PMBM outperforms the compared filters in the challenging scenario, while its multi-Bernoulli mixture parameterisation is more efficient than the δ-GLMB form.

APPENDIX A

Appendix A proves the relevant likelihood identity by induction over measurement and target-set components. The proof separately handles the cases of adding a measurement-associated component and a target component.

  • APPENDIX A: The appendix uses induction to prove the stated likelihood result.
  • APPENDIX A: The proof establishes equality between the likelihood forms l and l_s for finite measurement and target sets.
  • APPENDIX A: The induction step treats terms involving a new measurement through the corresponding likelihood factors and reduced sets.
  • APPENDIX A: A second induction case handles the addition of a target component and removes that component from the remaining target set.

APPENDIX B

Appendix B derives the update of a Poisson prior and proves the prediction result for an MBM representation. The predicted density remains an MBM when survival and birth components are convolved.

  • APPENDIX B: The appendix derives the Poisson-prior update yielding the parameters in equations (16)–(24).
  • APPENDIX B: The proof of equation (27) proceeds by induction, beginning with the zero-component case.
  • APPENDIX B: The induction separately handles an empty added component and a singleton target component.
  • APPENDIX B: For MBM births, the prediction combines the MBM density of surviving targets with the MBM density of new births using convolution.
  • APPENDIX B: The resulting predicted density corresponds to an MBM.

APPENDIX E

Appendix E establishes the equivalence between labelled MBM representations and δ-GLMB densities. A δ-GLMB global hypothesis maps to Bernoulli components with deterministic existence probabilities, making it a labelled MBM special case.

  • APPENDIX E: The appendix first writes a labelled MBM, including the labelled MBM01 case, in δ-GLMB form.
  • APPENDIX E: Both densities vanish for repeated labels or labels outside the label space, while distinct valid labels yield the equivalence.
  • APPENDIX E: The reverse direction writes a δ-GLMB density as a labelled MBM with the MBM01 parameterisation and normalized global-hypothesis weights.
  • APPENDIX E: A δ-GLMB global hypothesis is indexed by a hypothesis and label subset, with labels in the subset existing and the rest absent.
  • APPENDIX E: The corresponding Bernoulli existence probabilities are deterministic: 1 for included labels and 0 for excluded labels.
  • APPENDIX E: The δ-GLMB mixture components are equivalent to global hypotheses in an MBM01 parameterisation, whose target existence is deterministic.
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